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my site; wellness [continue reading this..] In bifurcation theory, a field within mathematics, a pitchfork bifurcation is a particular type of local bifurcation. Pitchfork bifurcations, like Hopf bifurcations have two types - supercritical or subcritical.

In continuous dynamical systems described by ODEs—i.e. flows—pitchfork bifurcations occur generically in systems with symmetry.

Supercritical case

File:Pitchfork bifurcation supercritical.svg

Supercritical case: solid lines represent stable points, while dotted line represents unstable one.

The normal form of the supercritical pitchfork bifurcation is

$\frac{dx}{dt}=rx-x^3.$

For negative values of $r$, there is one stable equilibrium at $x=0$. For $r>0$ there is an unstable equilibrium at $x=0$, and two stable equilibria at $x=\pm \sqrt{r}$.

Subcritical case

File:Pitchfork bifurcation subcritical.svg

Subcritical case: solid line represents stable point, while dotted lines represent unstable ones.

The normal form for the subcritical case is

$\frac{dx}{dt}=rx+x^3.$

In this case, for $r<0$ the equilibrium at $x=0$ is stable, and there are two unstable equilbria at $x=\pm \sqrt{-r}$. For $r>0$ the equilibrium at $x=0$ is unstable.

Formal definition

An ODE

$\dot{x}=f(x,r)$

described by a one parameter function $f(x,r)$ with $r\in \mathbb{ℝ}$ satisfying:

$-f(x,r)=f(-x,r)$  (f is an odd function),

$\begin{array}{lll}\frac{\partial f}{\partial x}(0,r_o)=0,& \frac{{\partial }^{2}f}{\partial x^2}(0,r_o)=0,& \frac{{\partial }^{3}f}{\partial x^3}(0,r_o)\ne 0,\\ \frac{\partial f}{\partial r}(0,r_o)=0,& \frac{{\partial }^{2}f}{\partial r\partial x}(0,r_o)\ne 0.\end{array}$

has a pitchfork bifurcation at $(x,r)=(0,r_o)$. The form of the pitchfork is given by the sign of the third derivative:

$\frac{{\partial }^{3}f}{\partial x^3}(0,r_o)\{\begin{array}{cc}<0,& \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}\\ >0,& \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}\end{array}$

References

• Steven Strogatz, "Non-linear Dynamics and Chaos: With applications to Physics, Biology, Chemistry and Engineering", Perseus Books, 2000.
• S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos", Springer-Verlag, 1990.

See also

• Bifurcation theory
• Bifurcation diagram